The Ensemble Quantum State of a Single Particle

The derivation of the statistical nature of the quantum mechanical wave function is presented within the formalism of quantum mechanics and the second quantization. The statistical wave function may be derived for non relativistic bosons, non relativistic fermions, and relativistic bosons by employing the commuting field operator . For relativistic electrons a strictly anticommuting must be employed to derive the statistical wave function (spinor). The discussion at the end of the paper aims to show the physical plausibility of a statistical wave function.     

Thermodynamic Potential of Relativistic Electron Plasma

Within the real time formalism of quantum field theory at finite temperatures based on the closed-time-path Green's function approach, a closed analytical expression of the thermodynamic potential of a relativistic electron plasma is derived under the random phase approximation by summing up all ring diagrams. The result is natural extension to the relativistic case sf our previous formula derived in the case of a Coulomb gas.     

Quantum field model with unrenormalizable interaction

The unitary relativistic model of quantum field theory with rapidly increasing spectral function (i.e. it grows faster than any finite power of momentum) is investigated. It is shown that there exist nontrivial Lagrangians, leading to this kind of spectral functions and allowing to construct the local theory without the ultraviolet divergences on their basis. In this theory theS-matrix is unitary and not equal identically to unity.     

Quantum statistical monte carlo methods and applications to spin systems

A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem⁽¹⁾ thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures. Invited talk presented at “Frontiers of Quantum Monte Carlo,” Los Alamos National Laboratory, September 3–6, 1985.     

Relativistic quantum coin tossing

A relativistic quantum exchange protocol enabling “coin tossing at a distance” between two participants is proposed. The exchange protocol is based on the fact that to distinguish a pair of orthogonal states with certainty in relativistic quantum mechanics requires a finite time that depends on the structure of the states themselves. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 10, 684–689 (25 November 1999)     

Differentiable probabilities: A new viewpoint on spin,gauge invariance,gauge fields,and relativistic quantum mechanics

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field but also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model.     

Relativistic quantum bit commitment in real-time

A relativistic quantum exchange protocol making it possible to implement a bit commitment scheme is realized. The protocol is based on the idea that in the relativistic case the propagation of a field into a region of space accessible for measurement requires, in contrast to the nonrelativistic case, a finite time that depends on the structure of the states. The protocol requires one classical and several quantum communication channels. It turns out that it is possible in principle to preserve the secret bit for as long a period of time desired and with probability as close to 1 as desired.     

A Classical and Quantum Relativistic Interacting Variable-Mass Model

A classical and quantum relativistic interacting particle formalism is revisited. A Hilbert space is achieved through the use of variable individual particle rest masses, but no c-number mass parameter is required for the relativistic free particle. Boosted center of momentum states feature in both the free and interacting model. The implications of a failure to impose simultaneity conditions at the classical level are explored. The implementation of these conditions at the quantum level leads to a finite uncertainty in interaction times, perhaps more closely modeling the exchange of virtual particles in quantum field theory. This work is compared and contrasted with other variable mass models in the literature.     

Hyperfunction Quantum Field Theory: Localized Fields Without Localized Test Functions

This note addresses the problem of localization in quantum field theory; more specifically we contribute to the ongoing discussion about the most appropriate concept of localization which one should use in relativistic quantum field theory: through localized test functions or through the fields directly without localized test functions. In standard quantum field theory, i.e., in relativistic quantum field theory in terms of tempered distributions according to Gårding and Wightman, this is done through localized test functions. In hyperfunction quantum field theory (HFQFT), i.e., relativistic quantum field theory in terms of Fourier hyperfunctions this is done through the fields themselves. In support of the second approach we show here that it has a much wider range of applicability. It can even be applied to relativistic quantum field theories which do not admit compactly supported test functions at all. In our construction of explicit models we rely on basic results from the theory of quasi-analytic functions.     

Relativistic theory of fermions and classical fields on a collocation lattice

We present a novel numerical method for solving dynamical strong field problems in quantum mechanics and classical field theory based on expansion of functions in terms of splines. The method differs from traditional approaches by the introduction of a mapping onto a collocation lattice, which is generally nonuniform and time dependent depending on the particular physical application. This approach results in a set of finite matrix transformations of a type which can be evaluated rapidly on supercomputers possessing either vector or matrix coprocessors. As an example of the method, we present a study of the relativistic quantum-mechanical many-electron problem interacting via very strong time-dependent classical fields.     

Quantum Fields Obtained from Convoluted Generalized White Noise Never Have Positive Metric

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (Lévy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by Baumann, based on the Dell’Antonio–Robinson–Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.     

Finite-Temperature Field Theory on the Light Front

The formulation of statistical physics using light-front quantization, instead of conventional equal-time boundary conditions, has important advantages for describing relativistic statistical systems, such as heavy ion collisions. We develop light-front field theory at finite temperature and density with special attention to quantum chromodynamics. First, we construct the most general form of the statistical operator allowed by the Poincaré algebra. In light-front quantization, the Green’s functions of a quark in a medium can be defined in terms of just two-component spinors and do not lead to doublers in the transverse directions. Since the theory is non-local along the light cone, we use causality arguments to construct a solution to the related zero-mode problem. A seminal property of light-front Green’s functions is that they are related to parton densities in coordinate space. Namely, the diagonal and off-diagonal parton distributions measured in hard scattering experiments can be interpreted as light-front density matrices.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

Some New Developments in the Theory of Path Integrals,with Applications to Quantum Theory

A survey of recent developments concerning rigorously defined infinite dimensional integrals, mainly of the type of “Feynman path integrals,” is given. Both the theory and its applications, especially in quantum theory, are presented. As for the theory, general results are discussed including the case of polynomially growing phase functions, which are handled by exploiting the connection with probabilistic functional integrals. Also applications to continuous measurement theory and the stochastic Schrödinger equation are given. Other applications of probabilistic methods in non relativistic quantum theory and in quantum field theory, and their relations with statistical mechanics, are discussed.     

Point-Form Hamiltonian Dynamics and Applications

This short review summarizes recent developments and results in connection with point-form dynamics of relativistic quantum systems. We discuss a Poincaré invariant multichannel formalism which describes particle production and annihilation via vertex interactions that are derived from field theoretical interaction densities. We sketch how this rather general formalism can be used to derive electromagnetic form factors of confined quark?Cantiquark systems. As a further application it is explained how the chiral constituent quark model leads to hadronic states that can be considered as bare hadrons dressed by meson loops. Within this approach hadron resonances acquire a finite (non-perturbative) decay width. We will also discuss the point-form dynamics of quantum fields. After recalling basic facts of the free-field case we will address some quantum field theoretical problems for which canonical quantization on a space?Ctime hyperboloid could be advantageous.     

The role of type III factors in quantum field theory

One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e. factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III, factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e. the way they are embedded into each other